Theorem hlatjidm

Description: Idempotence of join operation. Frequently-used special case of latjcom for atoms. (Contributed by NM, 15-Jul-2012)

	Ref	Expression
Hypotheses	hlatjcom.j	|- .\/ = ( join ` K )
	hlatjcom.a	|- A = ( Atoms ` K )
Assertion	hlatjidm	|- ( ( K e. HL /\ X e. A ) -> ( X .\/ X ) = X )

Proof

Step	Hyp	Ref	Expression
1		hlatjcom.j	|- .\/ = ( join ` K )
2		hlatjcom.a	|- A = ( Atoms ` K )
3		hllat	|- ( K e. HL -> K e. Lat )
4		eqid	|- ( Base ` K ) = ( Base ` K )
5	4 2	atbase	|- ( X e. A -> X e. ( Base ` K ) )
6	4 1	latjidm	|- ( ( K e. Lat /\ X e. ( Base ` K ) ) -> ( X .\/ X ) = X )
7	3 5 6	syl2an	|- ( ( K e. HL /\ X e. A ) -> ( X .\/ X ) = X )